Left-Right Symmetry and Neutrino Masses in a Non ... - Fermilab

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Left-Right. Symmetry and Neutrino Masses in a. Non-Perturbative. Unification. Framework. P.Q. Hung and S. Mohan'. Fermi National Accelerator Laboratory.
FERMILAB-PUB-87/212-T November,

Left-Right Symmetry Non-Perturbative

P.Q. Hung Fermi National

and Neutrino Masses Unification Framework

1987

in a

and S. Mohan’ Accelerator Laboratory

P. 0. Boz 500, Batavia, IL 60510

Abstract Within

the non-perturbative

unification

framework pioneered by Maiani,

Parisi and Petronzio, it is shown how the mass of the right-handed W boson can be determined just by knowing the values of sin’ 8, and ag(= gz/4lr) at the Fermi scale Ap u 250 GeV. Consequently, the knowledge of A4wR helps to determine the light Majorana neutrino mass which is my 2 O(l/Mw,). The best bounds we can obtain here are Mw, X 8 TeV, m,, s 5 x 10~eel’, rn,* S 0.2eV, rn”, s 65ev. lP.Q. Hung and S. Mohan on leave of absencefrom Department of Phyaies, University of Virginia, Charlottesville, VA 22901

-l-

The mystery

of parity

merous investigations its simplest

violation

Unified

can only be understood

extension,

example

broken

parity

symmetric

hand,

This idea is quite attractive

modelsl’l

and it would

on the value of AR. The Sum

breaking

to be AF = (&Gr)-i

for Aa except for the lower limits

has sparked nu-

model as well as in

violation

of the standard it is quite

at some scale, An, above which

of left-right

experimentally

i.e., SU(5),

On the other

interactions

In the standard

origin.

if one knows the origin

and its gauge interactions. spontaneously

in the electrowesk

into its possible

Grand

FERMILAB-Pub-871212-T

is intrinsic particle

possible

spectrum

that

it is restored.

of the type

sum

be very helpful

parity

This

SU(2)n

x

and is

is the x

U(1).

if one could get a handle

scale or the Fermi scale AF is determined

c1 250 GeV. No such determination on the mass of the SU(2)a

is found

gauge bosons, namely

MwR > 300 GeV or 1 TeV. Is it somehow possible to relate the two scales AF and An? This is the question would like to address in this letter.

It turns out that under a certain

assumptions,

does exist.

such a relationship

to calculate

sin20,(AF)

where An is with implications

and as(A,)

on neutrino

in this note was advanced by Maiani,

dard model131 are the near-infrared It means two things: U(l),)

x

are small;

asymptotically

free (U(l),

at which

perturbation

point

singularity).

that the “low”

At AF, as,

theory

couplings

sin* ~,(A.w) and as(Ap)

of standard

extra families no Grand

ingredient and os(A~)

of the stanfree

(of SU(3),

are themselves

non-

(this is the famous

Landau

to how large

enabled the authors

of Refs.

(Ref. [5] also applied

the same method

in the MPP scenario

is a relatively

In general, this number

to have reasonable Another

x

grow large at some scale Ao to be insensitive

are. These considerations

were given mass of O(4).

Unified

are found

quark and lepton families.

and 10 in order for sin’ e,(A,)

and SU(2)t

ceases to be valid

[2,4,5] to compute

A crucial

and ov(A.~)

is) and all couplings

couplings

sector).

has deep

of a non-asymptotically

or,

above AF, SU(3),

the “high”-energy

number

determine

Parisi, and Petronsio[21

energy couplings

stable fixed points

already

The “lawn-energy

the Yukawa

us

masses.

several years ago. It is the assumption

sum

in turn,

allow

respect to AF. It also turns out that this determination

The concept employed

theory.

set of reasonable

In fact, these assumptions

and such computations,

we

crucial

values.

element

gauge group was assumed in the computation

to

large

is between

8

Many of the

is the fact that

of sin2 E’,(A,).

FERMILAB-Pub-871212-T

-2-

The simplest

extension

of the standard

x SU(~)J, x SU(2)R

x U(l)‘.

model with,

the calculation

in mind,

the right-handed

breaking

The following

We now apply

set of fermion

of sin’ fl,(A,),

we have $i

= (3,2,1,

numbers.

right-handed

and the connection

2, -1)

N o t ice that,

B and L are the baryon

The set $L,R Iq forms a standard

family

Ref. [l].

of

For

while for the quarks

U(l)’

here, we identify

and lepton

the last entries

Since in this case Q = 2’s~ + 2’s~ + v, quantum

os(A~),

and I& = (l,l,

-1)

5) and $k = (3,1,2,5).

where

scheme to this extended

and scalar fields are chosen following

we have 4: = (1,2,1,

U(l),-,,

the MPP

is SU(3),

symmetry

scale AR to the Fermi scale AF.

the leptons, with

model with left-right

numbers

respectively.

denote the B - L

in tip,

where one now also has a

neutrino.

The Higgs fields include

the following

sets:

1. ~=(1,2,2,0),A~=(1,2,1,1),A~=(1,1,2,1)or 2. ~$=(1,2,2,0),A~=(1,3,1,2),A~=(l,1,3,2). The first set (4, AL, Aa) parable

gives only Dirac masses to the neutrinos

to those of the charged

the neutrino

leptons.

masses, one has to put Majorana

see-saw mechanism gauge invariance

to obtain

gives Dirac masses to the neutrinos masses through

this point

later in this paper.

seen that

This process, however,

As pointed through

the couplings

with

out by Ref.

the coupling

of

violates

of symmetry

breaking

[l], this set not only

with

AL and An.

the above set of Higgs fields whose number the pattern

the smallness

even if we had MV~VR with M 5 AR. The second

and natural.

Majorana

are com-

mass terms in by hand and use the

the mass eigenvalues.

and is unnatural

set is more interesting

With

In this case, to explain

which

4, but it also gives

We will

come back to

is left arbitrary,

is as follows:

it can be

SU(2)& x sum

x

The one-loop renormalization group U(l)&L -‘Aa SU(2)L x U(l), -+hF U(l).,. (RG) equations describing the evolution of the couplings above An are given by dQL,.Q _ T, *dt -- g1 &rI and, for AF 2 E 2 AR: by % = b2rc&., % = bya:, 2L,RQ:L,R, wtre

oi = gf/4x

Also, t = en(p2/M2). 6(A~)+n++4n),i)r~

and &i and or correspond The R.G. coefficients = I$(-22+6(A~)+n++4tZ),i;1

to U(l)s-L

and U(l),

respectively.

are given as &L = bra = A(-22

+

= &($+S(AL)+Z(AR)),by

=

-3-

&(y+f(Ar)+n+),

where n is the number

Higgs fields, 6(A)

and, in general,

= n~/2

$(A,)

when

= $(A,)

left-right

A~,R are doublets

= $5(A)

Since the theory

symmetry

n+ the number of 4(2,2,0) C-1 I-) implies 6 (An) = 6 (AR) =

and 6(An)

= 6(An)

is non-asymptotically

p-function.

Explicitly,

a, = g;/4x

with

one has %

+ 4n),Ca

and oioi

terms.

symmetry

breaking

couplings.

= &(76n

+ O(a:ar,

standard

o,yi(Ao),

What

and

model with

with

which appears in the solution

that all couplings

will be neglected

to the one-loop

so that o;r(A~)

neglect the the

as long as one does

mass much larger

follows

= 1, Z)), where

we will

approximation

means in what

to the QCD

one Higgs doublet,

scale. From here on, we will assume that assumption

is ex-

b3 and Ca are given

- 306). In what follows,

fermions

this statement

o,Tr(Ao) +b&%,

a:ai(i

The coefficients

In fact, for the standard

We now make the MPP scale Ao.

= 2%

the two loop contributions

results of Ref. (51 show that this is a reasonable not have superheavy

= 6(A)

free above AF and since as(&)

= bs(~: + C&

gf being Yukawa

by b3 = &(-33

of families,

= 3na when AL,R are triplets.

pected to be of order 0.1, one has to include

oio,

FERMILAB-Pub-871212-T

than

the SV(2)r

such is the case.

grow large at a common is simply

equation, u bitn%.

that

namely

the term cy,:‘(A~)

=

We can then readily

find sinz8,(Ap) The value of sin* e,(A,)

is given in Table

have used I..,.

= l/128.

down to U(l),

one has o;r(A~)

cz;l(An)

+ &(Fn

(-22 + S(Ar)

= (o”::))

+ 6(A,)

1 for various

Since, at AR, Su(2)n = a;i(An)

+ n+)en$

+ n4 + 4n) en”.AF Ao, n, n+ and 6(An).

x U(l),-,

+ &;‘(An).

-22 + 6 + 8 + 2n+ + yn

symmetry

values of oa which

dominates

using

$‘(A,)

=

>

fnk AF

- 67~;~. (AF)] / [--22

+6+z]>, (4

has been used, i.e.,

of AR are given in Table 1. Integrating

accommodate

is assumed to break Upon

C-1

The following

We

we obtain

-AR =ezp AC where left-right

(.l)

(-)

6 (AL)

the two-loop

= 6 (AR).

equation

Various

values

for oa, we obtain

the

are also shown in Table 1. results

emerge.

n = 8,9 standard

over the two-loop

term

The minimal families.

model with

n+ = na = 1 can only

For n 5 7, the one-loop

Cacr: for os(A~)

term

bsa$

N 0.1 and since, in this case,

FERMILAB-Pub-871212-T

-4-

b3 < O,SU(3),

is asymptotically

for n = 10, the minimal bound

and M,).

families

predicts

0.056 S os(A~)

The predictions

in the minimal

for os(A~)

model with

Ar,n

is 8 or 9. The same conclusion

(here 0.053 SZ os(A~) sin* Q,mi”(A,),

As

S 0.06 where the lower

c- 0.217 (these values of sin’ 9, are obtained

sin* @“(A,) say that,

model

to our assumption.

ma2( AR) N 0.24 and the upper to sin* 19~

corresponds

of Ml

free above AF contrary

bound

corresponds

to

from the measurements

are too small

being triplets,

and one can safely

the number

of standard

holds for the case when Ar,n

are doublets

S 0.056 for n = 10). One can lower a little

bit the value of

to say 0.21, with the effect that the upper bound on as

is altered

very little. Before we discuss the possible values of AR, it is important Ao are reasonable and os(A~).

since AR depends on Ao (Eq. (2)). The criteria

We have stated

above that

is how well does one know os(A~). finds os(M*) handle

firm

of Agoo

= &[l

+ 2?]

+ O(t-r)

mt, as result

can be computed is 0.09 ,S os(A~)

corrections

consensus

200 MeV.

there

with

of T-decay

For various mass

S

250 GeV. The

since the uncertainty

100 MeV to 300 MeV

(approximately)

and since there is also a possibility

fourth

mass

range for os(A~)

with

the following

As for sin’ e,(A,), With

model

(2) tells us that,

we will consider

in mind,

concerning

that

as a reasonable

An = Ao.

the predictions of families

As AC increases

value, An decreases. This is shown in Table 1 where the smallest

AR corresponds

to sin* B,(AF)

of the minimal

model

1.2 x 10” GeV, An CI 9

with x

10”

= 0.24. One can readily AL,R being

triplets

GeV, os(A~)

a

of

AF < AR < AC. In fact., eq.

model and for a fixed number

value of Ao for which

that

the range 0.21 S sin* Bw(AF) S 0.24.

we now first examine

AR. We require

can run from

.S 0.12.

we shall consider as reasonable

in the minimal

there is a minimum

250 GeV exists,

one 0.08 S as(A,)

the above constraints

the minimal

minimal

S

in Ag\

as, we values of

S 0.1. However,

generation

is no

is for the value

When we “run”

is flavor-dependent.

for three generations

to as, one

Measurements

= 200 f 50 MeVlsl.

9 AZ:

The question

Although

the general

in the &?S scheme to be around in which

the two-loop

where t = en&-.

on the precise value of Acon,

use the convention

will be sin’ B,(AF)

czs(A~) 5 0.06 is too small.

Including

give, on the average, the value A$ will

to see which values of

(8 or 9), from that value of

see that the best prediction

is given

by n = 8,Ao

= 0.117 and sin’ 0,(A,)

= A4r =

= 0.207. From

-5-

FERMILAB-Pub-871212-T

Table 1, one notices that when AL,R are doublets, of os(A,)

N 0.07 for sin* tJ,(AF)

for os(A~)

An = (3,1,2),

the B’s mass, left-right

the minimal

left-right

An = (1,3,2)

predicts

symmetry

implies

MuR = $gRAR N 5 x 10”

giving

discuss the neutrino can AR become?

too small a value

u 0.216 or a too small value of sin* &(AF)

u 0.117. In summary,

one 4 = (2,2,0),

one gets either

An c- 9

8 families

one has to go beyond

= gn N 1.11

MwR more, when we

GeV. We shall talk about of immediate

and

1O’l GeV. As for

x

gn = gn and, at An,g~

masses. The question

It is clear that

model with

Y 0.182

importance

is: how small

the minimal

model in order

to lower the value of An. The simplest

extension

of the minimal

more than one set of Higgs scalars.

model is the situation

We have looked at the following

na 2 2 and 2) na = 1, n+ 2 2. We first concentrate of the model in which imposed. With

The results

AL,R are triplets.

0.21 s

sin*B,(AF)

again only n = 8or9 gives acceptable 1.3

x

best results

lo’* Gel’, An = 1.4

na = 2,Ao

Again,

cases: 1) n+ =

on the more interesting

the requirement

version

AF < AR < Ao is

are shown in Table 1.

the constraints

have the following

in which there are

= An = 2

x x

S 0.24 and 0.08 .S os(A~)

results.

For case (1) with

S 0.12,

n+ = nA 2 2, we

(lowest values of AR): n = 8, n+ = nA = 2, Ao =

10’ Gel’, sin* e,(AF) 10”

GeV,sinZB,(AF)

= 0.24, aa

= 0.119; n = 9, ng =

= 0.227,as(A~)

= 0.08. For both

n = 8 and n = 9, n+ = nA 2 3 is unacceptable. For case (2) with by:

n = 8,nA

na

= 1,n4

=

1,ng

= 3,Ao

2, the best results

2

= 10 l9 GeV,AR

(lowest

An)

are given

= 1.1 x 10s GeV,sinZ8,(AF)

=

0.237, CQ(AF) = 0.117; n = 9, no = 1, n+ = 5, Ao = AR = 1.5x 10” GeV, sin’ &,,(AF) = 0.236, os(A~)

= 0.08. For n = 8 and 9, n+ 2 4 and nd > 6 are unacceptable

The less interesting and n = 8,Ao 0.117(~ 1.5 x 10”

case of doublet

= 1019 GeV,An

GeV,sin*

&,(A,)

AL,~ gives the best value for n+ = na = 3

= 8.8 x 10’s GeV,sin*

= na 2 4 is unacceptable).

respec-

6’,(AF)

= 0.23, and os(A~)

=

For n = 9, one has ng = na = 5,Ao = An =

= 0.241, os(A~)

= 0.08 (higher

values of n+ = na are

unacceptable). An examination

of the above results

of An one can get is around

and Table 1 reveals that

14 TeV. This correspond

the lowest value

to an interesting

lower bound

FERMILAB-Pub-871212-T

-6-

on Mw,,

namely

(Mw,

= igL,nhR):

MwR 2 8 Tel’ We would

like to stress here the fact that

solely, within

the MPP framework,

( 3)

the above lower bound

on Mw,

from sin* 0, and as. The implication

comes

on neutrino

masses is discussed next. Let us reiterate

the results

obtained

In the minimal

model with

ng =

with sin’ &,,(A,)

and

X 8 TeV. These are solely from sin* 0,(A F ) and cys(A~) regardless of how neutrino

In the non-minimal

the bounds

earlier.

MwR X 5 x 10” Gel’, consistent case (n+ = no = 2,n = ~),Mw,

na = 1 and n = 8 generations, os(A~).

obtained

masses come about. In the left-right

symmetric

through

the coupling

Yukawa

couplings

leptons;

and, if AL,n are triplets,

are of the form:

gLd$n,

= r&*rr)

Let us look specifically

Following

masses are obtained

gives < AL >=

and

for leptons only (C is the at the Yukawa couplings

+ $gCraAn$~n)

(v”,

These

for both quarks

Ref. [l] the most general couplings

+ hZ$L$$R + ihs($$CrrAr$L

tion of the potential[‘l

$r$$n($

~~CrZA~$~,~~Crz.An~~

matrix).

of a single lepton family, Ll, = hi$~#n

here, fermion

with the Higgs fields 4 and AL,R (if AL,R are triplets)l’].

Dirac charge-conjugation

Ic

model considered

$=

are given by

+ h.c.. The minimiza( ;R

$=

0

where, in general, K’ < IC (suppression of I?‘, - WR mixing), Vn >> K, and 0 K’ i 1 vL = ~rc*/vR < n(r is the ratio Higgs self-coupling). The charged lepton mass is given by mc- = hld+h2s

while the mass matrix

from (vTiVT)MC

+ h.c.(N

v

i N 1 a = hpL, b = --hpR,c = $(hln+ b > a, c gives the following

the limit formula

7